Related papers: Identities involving Narayana polynomials and Cata…
We answer the question in the title in the negative by providing four proofs.
In this paper, we consider an analogue of Catalan polynomials and give some identities of symmetry for those polynomials by using fermionic $p$-adic integrals on the ring of $p$-adic integers
In this paper, we connect two well established theories, the Fibonacci numbers and the Jordan algebras. We give a series of matrices, from literature, used to obtain recurrence relations of second-order and polynomial sequences. We also…
In this paper, we focus on Narayana numbers which can be written as a products of four repdigits in base $g$, where $g$ is an integer with $g\geq2$. We prove that for $g$ between $2$ and $12$, there are finitely many of these numbers.…
A sequence of coefficients appearing in a recurrence for the Narayana polynomials is generalized. The coefficients are given a probabilistic interpretation in terms of beta distributed random variables. The recurrence established by M.…
When searching for Calabi.Yau differential equations, often different formulas for the coefficients give the same differential equation. The coefficients are usually sums (simple, double or triple) of products of binomial coefficients. This…
We derive an identity involving Horadam numbers. Numerous new identities as well as those found in the existing literature are subsumed in this single identity.
We provide some variations on the Greene-Krammer's identity which involve q-Catalan numbers. Our method reveals a curious analogy between these new identities and some congruences modulo a prime.
In this paper, we show that there are only finitely many Narayana's numbers which can be written as product of three repdigits in base $g$ with $g \geq 2$. Moreover, for $2 \leq g \leq 10$, we determine all these numbers.
In this paper, we study some symmetric identities of q-Euler numbers and polynomials. From these properties, we derive several identities of q-Euler numbers and polynomials.
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
We study the modular resolution method using new tools called polynumbers and polyseries, introduced by Prof. Wildberger N.J. We try to prove an equivalence theorem of the existence and the uniqueness of the solutions of the modular…
We present a new alternating convolution formula for the super Catalan numbers which arises as a generalization of two known binomial identities. We prove a generalization of this formula by using auxiliary sums, recurrence relations, and…
In this article we prove some identities which allow us to evaluate some multiple unit square integrals. In our examples we will give the value of some double and triple integrals. Then, we prove several classical integral formulas with the…
In previous work on Clebsch-Gordan coefficients, certain remarkable hexagonal arrays of integers are constructed that display behaviors found in Pascal's Triangle. We explain these behaviors further using the binomial transform and discrete…
In this paper, we give several matrix representations for the Horadam quaternions. We derive several identities related to these quaternions by using the matrix method. Since quaternion multiplication is not commutative, some of our results…
In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. From our study of non- linear differential equations, we derive some new and explicit identities for Legendre…
We offer several new summation identities involving harmonic numbers, odd harmonic numbers, and Fibonacci numbers. Our results are derived using three different approaches: partial summation, polynomial identities and binomial…
Many classical $q$-series identities, such as the Rogers--Ramanujan identities, yield combinatorial interpretations in terms of integer partitions. Here we consider algebraically manipulating some of the classical $q$-series to yield…
The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a…